Influence of Thermal Residual Stresses on the Interface Crack

Transcription

1 Finite Elements in Engineering Applications 1992, p , INTES GmbH Stuttgart Influence of Thermal Residual Stresses on the Interface Crack S. Schmauder, M. Meyer Max-Planck-Institut für Metallforschung, Institut für Werkstoffwissenschaft, Seestraße 92, D-7000 Stuttgart 1, Germany Abstract In this paper, the influence of thermal residual stresses on the propagation of a crack along a material's interface is examined using the finite element method. Helen's virtual crack extension method is modified in order to calculate the energy release rate of the crack. The single mode stress intensity factors are separated by applying two different, specially chosen displacement fields and studying carefully their interaction by means of a J-like path independent integral. When the bonded phases are assumed to behave elastieally isotropie, correction functions as weil as local phase angles can be characterized by the two Dundurs' parameters. The striking feature of thermal residual stresses is their strong mode 11contribution at the interface crack tip. It is found that thermal stress intensity factors are a linear function of crack length while the resulting local phase angle is a linear function of Dundurs' first parameter. In the framework of linear elastieity thermal stress intensity factors may be superimposed on the stress intensity factors from applied loads such that a wide range of applications for composites is expected. Finally, a modified compact-tension-shear loading deviee for testing bimaterials is introduced. The corresponding calibration functions and local phase angles for this geometry were calculated and are discussed with respect to systematic variations of elastic properties, crack length ratios and applied mixed-mode loadings. 1. Introduction Composites develop thermal residual stresses during cooling from processing temperatures to room temperature due to relatively large thermal expansion mismatch. Bimaterials are, therefore, often prone to interfacial failure. Partieu

2 2 larly, in the case of brittle components thermal residual stresses should not be neglected. They may contribute strongly to the energy release rate of interface cracks, thus reducing the apparent toughness of bimaterials [1]. Cracks in isotropie, homogeneous materials tend to grow in opening mode; hence fracture toughness is characterized by a single mode parameter, the fracture toughness KIc. The fracture toughness of interface cracks in general is no longer a single mode stress intensity factor (SIF) but is a function of local mixed-mode conditions: Kc = f(ki, Ku). Interface cracks in perfectly brittle bimaterials (e.g.: glass/ glass) often tend to kink out of their initial plane, even if pure mode I loading is applied remotely. Cracks in more compliant bimaterials, e.g. in metal! ceramic composites, tend to grow along the interface regardless of minor local mode mixity. For these reasons it is necessary to quantify the local mode mixity at the crack tip as a function of elastic properties, crack length ratios and loading states. In particular, it is necessary to characterize the influence of thermal residual stresses on interface cracking in a quantitative manner. In this paper, an extension of an existing virtual crack extension based method for obtaining thermal stress intensity factors (TSIFs) will be described. Furthermore, preliminary results of a systematic study on TSIFs of interface cracks are presented and discussed. Moreover, stress' intensity factors (SIFs) for a compact-tension-shear loading device are calculated using the finite element method. 2. Characterization of Bimaterials Solutions for interface crack problems depend elastically only on the two dimensionless Dundurs' parameters a and ß [2] which are contractions of the four elastie constants (Young's moduli Ei and Poisson's ratios Vi) according to k(kl + 1) - (K2 + 1) a=k(kl + 1) + (K2 + 1) k(kl-1) - (K2-1) ß=k(Kl + 1) + (K2 + 1) (la) (lb)

3 3 where k = ~2/~.l1 is the ratio of shear moduli of both materials, Ki = 3-4vi for plane strain and Ki = (3-Vi) / (1+Vi) for plane stress is Muskhelishwili's constant [3] and the index (i = 1, 2) refers to material "1"and "2", respectively. The elastic constants of any material combination were shown to be limited by the following parallelograms [4] for plane strain and plane stress, respectively -1.0 ~ a ~1.0 a a ~ ß ~ a ~ ß ~ 8a (plane strain) (plane stress) (2a) (2b-1) (2b-2) As the sign of a and ß is reverted when materials are exchanged it is sufficient to examine only one half of the parallelogram given through eqns. (2), e.g.: a ~0.0. pa The following simple relationship has been found to exist between Dundurs' rameter a and Young's moduli of involved phases [5] (3) In eqn. (3), E7 is given by (plane strain) (plane stress) (4) and Ei are Young's moduli of the components (i = 1, 2). Thus, a is easily obtained using the ratio of the plane moduli E; and E; (Fig. 1). Since their discovery, Dundurs' parameters a and ß have often proven their usefulness in characterizing elastic properties of bimaterials and other material joints [1, 4, 6]. Because of the importance of Young's moduli, a is of primary

4 a Fig.1: Dundurs' first parameter, a, as a function of Young's moduli. interest in studies on the elastic response of composites. The dependency of a in eqn. (3) suggests that in most real material combinations with a ratio of (5) a is limited within lai ~ 0.6 (6)

5 5 0.5 ß ~ ~--- 01" _0'; o' 0.1.,~\OO O.O~ 11 ol IM ljil Q o AI / Ceramic. Mo.W. B.C o Cu/ Ceramic.Mo.w,B.C D. Ni / Ceramic,rvo.W,B.C Ti,Zr,Hf.V. Nb /Ceramic,Mo.w Ta.Cr.rvo.w /Ceramic '-44 / Plastics /Others Fig.2: Distribution of typical material combinations in the a-ß-diagram (plane strain). This hypothesis is supported by a systematic investigation on different material joints in [6, 7] as shown exemplarily in Fig. 2. It has been noted that nearly all technically relevant material combinations follow the relationship [8] (plane strain) (7) In agreement with eqn. (2b-l) the relationship [5, 9] (plane strain) (8) was introduced in order to reduce the number of parameters to a single one. It is worth mentioning that for identical Poisson's ratios the correlation between a

6 6 and ß given through eqn. (8) is exact. It holds also approximately for many practically important composites of components with une qual Poisson's ratios. In studies on practical implications of Dundurs' second parameter, ß was even set to zero arguing that ß is small according to eqns. (6) and (8) [10].Except for the case of identical Poisson's ratios the correlation between ß and elastic constants of involved phases remained unclear since. The relationship between ß and the elastic constants of involved materials (eqn. 1b) provides the basis for resolving these uncertainties and for correlating a and ß in a predictive sense for arbitrary Poisson's ratios. The relationship in eqn. (lb) between the four elastic constants and ß can be rewritten as (e.g. [11] for plane strain deformation) E---E v v2 2 1-v v2 + + El + E2 (plane strain) (9a) (plane stress) (9b) For identical Poisson's ratios (v = v1 = v2) it follows from eqns. (3) and (9) that 1 (l - 2v) ß="2 (l-v) a (plane strain) (loa) (plane stress) (lob) Eqn. (8) is re-discouvered from eqn. (loa) by setting Poisson's ratio to v = 1/3. However, it is also seen that depending on v other linear relations between a and ß can be achieved. Under plane stress assumptions a similar relationship holds for v = 1/3 (e.g.[12])

7 7 (plane stress) (11) In case of identical Poisson's ratios v = VI = v2 ~ 0.25 and for positive a values which fulfill eqn. (6) the above result (eqn. 10a) leads to a narrow range of possible ß values o ~ ß ~ 0.2 (plane strain) (12) plane strain v2=0.4 v2=0.3 v2= Fig.3: Dundurs' second parameter, ß, as a function of elastic constants (plane strain). The more realistic case is obtained from eqn. (9a) and can be written as

8 8 ß 4 1-v =~[(1-2Vl 1 1-v 1-v 1-v + 1-2V2]a+(1-2V1_ V2]1] 2 (plane strain) (13a) (plane stress) (13b) Two unexpected results can be noted: First, the range of possible ß values is limited when a and Poisson's ratio of the second material are fixed; and second, there exists a linear relationship between a and ß for any given ratio of Poisson's ratios. It is worth mentioning that eqns. (13) reduce to eqns. (10) when both materials possess the same Poisson's ratio. These results provide a simple way of predicting Dundurs' second parameter ß for technically important material combinations with Poisson's ratios in the range of 0.2 ~ v1' v2 ~ 0.4 (Fig. 3). 3. Problem Formulation In general, interface crack tips experience both normal and shear loading due to the elastic mismatch of adjoint materials in the presence of thermal residual stresses or under extern al loading. It has been noted that remote mode I loading produces predominantly crack tip opening while thermal residual stresses provide mainly shear contributions at the crack tip [6]. Using the notations given in Fig. 4, the near tip stress field für an interface crack lying between two infinite, homogeneous and isotropie materials can be described by [13] G" 1) = --J 2m 1 [Re(KriE)f!(S,E) 1) + Im(KriE)f!!(S,E)] 1) + TÖ'lÖ'l 1 ) (i, j = 1, 2) (14) where r is the radial distance from the crack tip, and the bimaterial constant E is a function of ß only E=2rc1 (l+ß) 1-ß (15) The dimensionless angular functions f!(s, 1) E) and f~~(s,e) 1) are related to the tractions across the interface and are given in reference [13]

9 9 CD Fig.4: Analyzed geometry of abimaterial with an interface crack. The non-singular T-stress of the all-component plays a major role for kinked.- crack segments [1, 14] and was recently identified [15] to be one half of the nominal residual stresses, ares, and given as (16) The difference in thermal expansion coefficients is obtained from Hooke's law as (plane strain) (plane stress) (17) In addition, the cooling interval is determined by the difference between room temperature Ta and processing temperature Tp i1t= Ta - Tp (18)

10 10 and E* is a mean Youngs' modulus of the bimaterial and defined as [16] (19) The local stress Held at the tip of an interface crack is produced by residual stresses as weil as applied stresses. The geometry functions f:j(8, E) are scaled in such a manner that the local stress Held ahead of the interface crack (8 = 0) is given by ) (20) The complex stress intensity factor K in eqn. (14) and eqn. (20) has the generic form (21) a is the crack length and P is the representative stress amplitude. By definition '" is the phase of KaiEwhere '" can be interpreted as the phase of the tractions at r = a, assuming that eqn. (21) still holds at this distance ahead of the crack tip. Yk is a dimensionless geometrie function of material properties, loading conditions and crack length ratios. According to Rice [17] aglobai SIF for interface cracks may, therefore, be defined in the usual manner, if the radial distance from the crack tip, r, is chosen as a fixed length quantity, r = ~. Due to this substitution we can rearrange eqn. (21) to ". "" r IE. Kr IE= KI(r )+ ikn(r ) = Yk P ~ ("J' a el'l' (22) where (23a)

11 11 (23b) These global SIFs Ki(~) (i = I, II) have the usual dimensions (MPa"'J'm)and may be interpreted in the conventional manner according to eqn. (23). A detailed discussion of stress oscillation and the effect of length quantities can be found in [17]. For our subsequent treatment of thermal stress intensity factors we incorporate at this point Ki(~ ) == Ki (i = I, II) and ~ = a. The fact that ~ = a lies obviously outside A the zone of K-dominance is of no consequence as long as r is recorded along with the results of \jf and as long as one is familiar with the \jf-transformation given through (24) Interface toughness values Kc are usually calibrated by tabulated y~pp-functions, which are given for a wide range of test geometries. As these kinds of calibrations do not take into account inherent stresses their application will lead to different Kc values for bimaterials with identical elastie properties and crack length ratios but different residual stress states. However, by definition Kc must be taken as a material characterizing parameter. A correct K calibration can be performed by superimposing single mode SIFs due to applied loads (app) and residual stresses (res) according to K. _ ICPP K;es (i = 1,2) (25) In analogy to the SIF definiton in the case of applied loads (equation 21), we may now express the complex TSIF through the nominal residual stresses using the relation K res = yres k O''V res _r- rca a-ie eill/res T (26) where yrs is the dimensionless geometrie function of Dundurs' parameters a and ß as weil as normalized crack length a/w. Thus, the thermal calibration of a

12 12 bi material interface crack geometry is reduced to determining y~es and ",res for the interesting range of material combinations and crack length ratios. The main objective of the following sections is, therefore, to present a new method to calculate thermal calibration functions y~es for arbitrary material combinations and crack length ratios. 4. Method An effective numerical method for calculating SIFs Ki (i = 1, 2) for interface cracks is given by Matos et al. [18]. This procedure is based on the evaluation of the J-integral of Rice [19]. using the finite element method. If there are two different displacement fields ua and ub, representing solutions to two different boundary value problems for the crack, the values of the energy release rate G associated with them are Ga and Gb. When the displacement fields ua and ub are summed to give uc, the value of G resulting is (27) where MI is a J-like path independent integral [20], characterizing the enrgetic interaction of the two displacement fields. Using eqn. (27) and the following relation between energy release rate and stress intensities [16] (28) the interaction energy can be expressed as (29) From this equation, the Kla (K2a) -SIFs can easily be extracted, if special Kb values are chosen, Le. KIb = L\KIband K2b = 0 (K2b = L\K2b and KIb = 0)

13 13 CD CD Fig.5 5chematic for calculating the effective displacement Held fun}. Following Hellen [21], the energy release rate G can be evaluated numerically by a virtual crack extension method (VCE) (30) where U is the potential energy of the body, a is the crack length, fun}contains the element displacement vectors and [5] is the stiffness matrix for the mesh of elements used to solve the problem. To apply the VCE-method for thermal stress problems, eqn. (30) must be modified since the initial strains due to the temperature difference have to be taken into account, additionally. The effective displacement Held fun}for use in eqn. (30) is then given by (31) where {un}act is the actual displacement Held of the cracked body and {un}iniis the initial displacmeent Held of the uncracked structure. Two finite element calculations are therefore necessary to provide {un}actand {un}ini(fig. 5)

14 14 rigid dislorled Fig.6: A typical ring of elements around the crack tip to be distorted in the J ca1cula tions. In a post-processing step the crack is virtually extended by rigidly moving a core of elements around the tip in order to determine a new stiffness matrix [5] (Fig. 6). then only the distorted ring of elements contributes with non-zero {as/ dal values to the energy release rate G in eqn. (30). Now a second displacments field uh is superimposed on uh == {unh for which KI =.1KI and K2 = 0, Le., (32) where fl is given in [17] and i = 1, 2 is again the material index. The ca1culation in eqn. (30) is now repeated with the vector {unl + {.1unh used instead of {unl alone. The result of this calculation, G +.1GI, can be shown from eqns. (27), (28) and (29) to be such that

15 15 (33) To obtain K2 the procedure can be repeated for an added vector uh == that Kl = 0 and K2 = ~K2 using {unh such ~u2 i = ~K2 21li -\I...r;- 2iC (1 + e1le e2m:) f2(r,8, E, K) (34) or alternatively using eqn. (28) to determine K2' A more detailed description of computing interfacial stress intesities can be found in Matos et al. [18]. 5. Thermal Stress Intensity Factors of Interface Cracks 0.30 a/_ a/_=0.2 a/_= a/_ I 0.0 I 0.2 I 0.4 a I 0.6 a/_=o.o Fig.7: Correction function for TSIFs as a function of crack length and elastic mismatch

16 16 In the following, results are presented for the geometry function of thermal resid 1.. f yres d h d' d" res f ua stress mtenslty actors k an t e correspon mg mo e mlxlty '" or use in eqns. (25). The dependence of geometry function Ykes on a is shown in Fig. 7. Obviously, Ykes is a decreasing function of crack length but increases with increasing a and ß values. For short cracks (a/w ::::: 0.1) there is a distinct effect of the elastic mismatch a, and ß on y~es, whereas long interface cracks (a/w ::::: 0.5) produce a modest change in this geometry function. 8/,.=0.5 8/W=0.4 8/w=0.3 8/,,-0.2 8/"=0.1 8/w=0.1 8/w=0.2 8/"=0.3 8/W=0.4 8/w=0.5 Fig.8: Local phase angle at the interface crack tip due to thermal residual stress as a function of crack length and elastic mismatch

17 17 Similarly, the local phase angle ",res develops significant mode I fractions for short cracks and increasing a and ß values. For the interesting range of moderate a values (a $ 0.5, Fig. 2) and typical notch lengths of 0.2 $ a/w $ 0.5 residual thermal stresses produce nearly a pure shear loading at the tip of the interface crack (Fig.8). For engineering purposes the following approximations may be used for 0.1 $ a/w $ 0.5 and ß = a/4, res 2 2 a Yk = { a }- { a } - (35a) w \j1es = 90- { : (:) (: )} a (35b) It is worth mentioning that the mode mixity is fully determined by the elastic mismatch and the crack length: the phase angle ",res is independent of the thermal mismatch, L1a, and the cooling interval, L1T. Only the absolute amount of Kres changes with ares = E*L1aL1Tfollowing eqn. (26). In the presence of applied stresses, e.g. when testing the toughness of real bimaterials, TSIFs can significantly increase the mode mixity and the effective SIF at the tip of interface cracks. 6. Calibrating the Compact- Tension-Shear Loading Device for Testing Bimaterials The geometry to be used for testing bimaterials is shown in Fig. 9. A similar compact-tension-shear loading device has been used successfully by Richard [22] to measure the fracture toughness in homogeneous materials. The mode mixity is varied by changing the loading angle ro. This angle can range from -90 to +90. For homogeneous materials pure mode I conditions are achieved when ro = 0 and pure mode II conditions when ro = ~900. For testing small bimaterial specimens having a low interface fracture toughness, the CTS loading device must be modified for eliminating inherent mass moments. Depending on the material to be tested, the resulting moment might cause SIFs which are dose to the critical K value of the bimaterial. For this ree

18 18 ason the original loading device is counterbalanced by additional compensation weights as shown in Fig. 9 [22]. F loading angle c.j / / 1) crs loading device 2) CTS specimen 3) counterbalance unit Fig.9: Modified CTS-loading device and specimen geometry. The finite element model used to solve the elastic boundary value problem is shown in Fig. 10. The model employs 8-noded quadrilateral plane membrane elements, with a square focused crack tip mesh. The loading device itself does not need to be modeled and can be simulated by the tabulated static/kinematic boundary conditions. These boundary conditions correspond to the loading angle (J) and are tabulated in Table

19 19 x Fig.l0: Finite element model of the CTS specimen and its loading grip. (J) -0.07*F -0.35*F -0.62*F -0.84*F -1.00*F -0.26*F O.OO*F 0.55*F 0.22*F d'e -0.97*F -1.00*F df Fy -0.50*F *F Fx 0.93*F 1.12*F 1.06*F 0.74*F 1.09*F - - Table 1: Static boundary conditions according to [23] for computing the CTS geometry using the finite element method

20 20 Increasing crack length ratios were simulated by moving the square focused crack tip mesh through the remaining structure. This provides crack length ratios of a/w=i/6 (i= 1, 2,3,4) (36) The specimen has been calibrated systematically für these crack length ratios and for a wide range of material combinations using the numerical procedure described in seetion 4. Beside this variation we utilized the characteristic bimaterial property by which most technically important material combinations can be described through eqns. (7) and (8). The K calibration of the CTS geometry following eqn. (22) uses a representative stress magnitude of F P=wt (37) where F is the applied load, w the specimen width and t the specimen thickness. The calibration of the CTS specimen is reduced to determining Yk and '" for the interesting range of material combinations, crack length ratios and loading angles [22]. The dependence of Yk and '" on a, ß = a/4, a/w and Cl) is shown in Figs. 11 and 12 (a-d). However, similar results have been obtained for different ß values according to the examined range of ß (equation 7). Obviously, the computed Yk and '" values are no longer symmetrie with respect to the sign of (J), except for the homogeneous case, a = ß = O. There is a significant difference relative to the resulting Yk and '" values whether the applied load is positively «(J) > 0 ) or negatively (Cl) < 0 ) introduced. In general we ascertain, that (38a) (39b)

21 21 holds for systems with a, ß #; O. Fig.11: CTS calibration functions, Yk = Yk (a/w, 00, a, ß). From Fig. 11 we also obtain an increasing crack length effect on this Yk asymmetry: A crack length ratio a/w = 4/6 needs a K calibration of Yk = 2.7 for 00 = -60 whilst (0 = +60 must be corrected with Yk = 3.3, considering a = 0.6. The local phase angle 'V (Fig. 12) is affected by changes in the crack length ratio. Short cracks with a/ w = 1/6 produce a phase angle of I'V I (00 = 1-90 )::::90 independent of the chosen a value. Therefore, mode II fracture result in mode II crack tip conditions, which are unaffected by the elastic mismatch if the crack length ratio is about a/w = 1/6. The a dependence becomes more evident when changing towards mode I fracture tests: At a loading angle 00 = 0, short cracks of a/w = 1/6 experience mixed-mode conditions with a small mode II contribution resulting in small 'V values of less than 10. Similar results were found by O'Dowd et al. [24] for bend bar geometries

22 GoIb -30 lt So !. _40-75' -1004!. c Fig.12: d CTS phase angles, \jf = \jf (a/w, 00, a, ß), for different a values and four different crack lengths (a-d). The functional effect of elastic mismatch a and ß on the phase \jf is somewhat different when passing over to crack length ratios larger that a/ w > 1/6: Increasing a values produce an increasing phase shift towards higher \jf values along the full range of loading angles. This entails negative KI values (\jf > 90 ) when remotely

23 23 loading in negative mode 11direction (w = -90 ), Le. the crack surfaces experience compressive stresses. Positive w values always lead to a crack opening mode, independent of the elastic mismatch and crack length ratio. 7. Conclusion A modified virtual crack extension technique for evaluating thermal stress intensity factors in interfacial fracture has been presented. In summary, it may be concluded that the stress intensity factors and geometry functions of thermally stressed bimaterials with interface cracks have been presented in a systematic manner as a function of crack length and elastic mismatch. Simple analytical express ions of these functions were presented. It was found that residual thermal stress result in a dominating shear component at the interface crack tip. The presented results on calibration functions and mode mix for the CTS specimen predict a strong influence of the elastic mismatch on the onset of crack growth as weil as on the preferred kinking angles for cracks deviating from the interface, on the condition that interfacial stress intensity factors can be taken as crack controlling parameters. There is strong evidence that careful attention must be directed on the sign of Ku values when examining bimaterials with E "# O. The computation of local mode mixity was performed using a modified virtual crack extension technique. Asymmetrie trends in the calibration functions Yk and in the phase angle 'JI were found with respect to the sign of the loading angle w. Increasing crack length ratios intensify the influence of elastic misfit properties on Yk and 'JI. In summary, it can be concluded that the stress intensity factors and phase angles of the CTS bimaterial specimen have been presented for the first time in a systematic manner as a function of crack length, elastic mismatch and loading state. The exact knowledge of these SIFs is important for all systems with elastic and thermal misfit. In the framework of linear elasticity the single SIFs Ki(i = 1, 2) may be superimposed on stress intensity factors due to thermal stresses in order to obtain effective SIFs acting in thermally prestressed bimaterials

24 24 Acknowledgement This work was supported by the German Research Foundation (DFG project EI 53/13-1); the support is gratefully acknowledged. References [1] G. Dreier, S. Schmauder, G. Eissner, "Propagation and Fracture Energy of Interface Cracks in Elastically Similar Brittle Materials Under Mixed Mode Loading Conditions", Int. J. Fatigue Fract. Eng. Mater. 5truct. (1992) in press. [2] J.W. Dundurs, "Discussion, Edge-Bonded Dissimilar Orthogonal Elastic Wedges Under Normal and Shear Loading", ]. Appl. Mech.36 (1969) [3] N.!. Muskelishwili, "Some Basic Problems of the Mathematical Theory of Elasticity", transl. by J.R.M. Radok, R. NoordhoH Publ. Company, Groningen, Netherlands (1953). [4] D.B. Bogy, "On the Problem of Edge-Bonded Elastic Quarter-Planes Loaded at the Boundary", Int. J. Sol. 5truct. 6 (1979) [5] S. Schmauder, "Theory of the Elastic Interface Crack (in German)" in: CFI (Ceramie Forum International) 2 (1987) [6] T. Suga, "Fracture Mechanical Characterization and Determination of the Bonding Strength for Material Joints (in German)", Ph.D. Thesis, University of Stuttgart (1983). [7] T. Suga, G. Eissner and S. Schmauder, "Composite Parameters and Mechanical Compatibility of Material Joints," J. Camp. Mat.22 (1988) [8] M. Meyer, S. Schmauder, " Thermal Stress-Intensity Factors of Interface Cracks", Int. J. Fract. (1992) submitted. [9] S. Schmauder, M. Meyer, "Correlation Between Dundurs' Parameters and Elastic Constants", Zeitschrift für Metallkunde (1992) submitted

25 25 [10] J.W. Hutchinson, "Mixed Mode Fracture Mechanics of Interfaces", in: Metal-Ceramic Interfaces, Acta 5cripta Metallurgica Proceedings, Series 4, Eds.: M. Rühle, A.G. Evans, M.F. Ashby, J.P. Hirth, Pergamon Press, Oxford, England (1990) [11] H.M. Jensen "On the Blister Test for Interface Toughness Measurement", Eng. Fract. Mech. (1992) submitted. [12] H.C. Cao and A.G: Evans, "On Crack Extension in Ductile/Brittle Laminates", Acta metall. mater. 39 (1991) [13] J.R Rice, Z. Suo and J.-S. Wang, "Mechanics and Thermodynamics of Brittle Interfacial Failure in Bimaterial Systems", in: Metal-Ceramic Interfaces, Acta 5cripta Metallurgica Proceedings, Series 4, Eds.: M. Rühle, A.G. Evans, M.F. Ashby, J.P. Hirth, Pergamon Press, Oxford, England (1990) [14] M.-Y. He, A. Bartlett, A.G. Evans, J.W. Hutchinson, "Kinking of a Crack out of an Interface: Role of ln-plane Stress", J. Am Ceram. 50c. 74 (1991) [15] G. Dreier, S. Schmauder, M. Meyer, "T-Stress due to Thermal Residual Stresses in Bimaterials with Isotropic, Linear Elastic Components", Eng. Fract. Mech. (1992) to be submitted. [16] D.R Mulville, P.W. Mast and RN. Vaishnav, "Strain Energy Release Rate for Interfacial Cracks between Dissimilar Media", Eng. Fract. Mech. 8 (1976) [17] J.R Rice, "Elastic Fracture Mechanics Concepts for Interfacial Cracks", J. Appl. Mech. 55 (1988) [18] P.P. Matos, RM. McMeeking, P.G. Charalambides and M.D. Drory, "A Method for Calculating Stress Intensity Factors in Bimaterial Fracture", Int. J. Fract. 40 (1989) [19] J.R. Rice, "A Path-Independent and Approximate Analysis of Strain Concentration by Notches and Cracks", J. Appl. Mech. 35 (1968) [20] F.H.K. Chen and RT. Shield, "Conservation Laws in Elasticity of the J- Integral Type", ZAMM 28 (1977) [21] T.K. Hellen, "On the Method of Virtual Crack Extensions", Int. J. Num. Meth. Eng. 9 (1975) [22] M. Meyer, S. Schmauder, G. Eissner, "Mixed-Mode Fracture Investigations of Interface Cracks in Dissimilar Media", Int. J. Fatigue Fract. Eng. Mater. 5truct. (1992) in press

26 26 [23] H.A. Richard, "Bruchvorhersagen bei überlagerter Norrnal- und Schubbeanspruchung von Rissen", VDI-Forschungsheft 631, VDI-Verlag, Düsseldorf (1985). [24] N.P. O'Dowd, C.F. Shih and M.G. Stout, "Test Geornetries for Measuring Interfacial Fracture Toughness", Int. J. Solids Structures 29 (1992)